SBG
You are right to question how up to date NATO exercise planners are, I wonder about it myself when I hear some of the mumbo jumbo that they come up with. However, I don't think that invading the eastern US coast was ever a Cold War scenario, it merely represents lots of other places in the world. The exercise will not be limited to North Carolina but will simulate a larger theatre of operations with different task groups doing different tasks.
During the actual landing stage of an amphibious assault, the troops will be put ashore by various means, which need to be protected. See
here.
Beaches vary in gradient, water depth and other characteristics. A beach may have a steep gradient, with the depth of the water increasing rapidly with distance from the shore. Alternatively it may be shallow, so that the frigates escorting the landing craft have to remain several miles off shore, leaving the landing forces in a very vulnerable position, outside the range where our frigates can provide extended point defence. History demonstrates how dangerous even just a few enemy aircraft are during a landing.
Also a task group operating some distance from other friendly forces (leaving aside unilateral operations) really need to be able to provide their own air defence, particularly if other friendly forces are several hundred miles away, and the enemy forces are near - very likely when operating in the littoral.
Having the Sea Harrier and organic air defence allows the United Kingdom to deploy an autonomous Carrier Task Group and/or Amphibious Task Group which can defend itself from air, surface and submarine threats, perform mines countermeasures operations if needed, put forces ashore and then provide aviation, logistics and naval gunfire support. It might interest you to know that new ammunition, with a longer range, is being introduced for the 4.5 inch gun. Basing logistics and aviation support at sea is a concept that was proved in the Gulf last year, as noted by the MOD report on Operation
Telic.
Needing US air cover (possibly needing AAR support) would reduce the autonomy, and hence the degree of influence that the UK has. Being able to provide useful assets has given the UK influence over Washington before, and the sort of capability I have just described is likely to be seen as more useful than providing a few more aircraft to go with the hundreds of their own. The same might be said about Tomahawk.
Up until now, everyone has concentrated on warfighting operations. However, we should also consider a number of possible operations other than war, which may be more likely to occur without US involvement. These might include...
Peace support operations
Low intensity intervention (eg Sierra Leone - see note below)
Evacuation of British (and possibly other) civilians from a war zone
Humanitarian operations
Long range hostage rescue (where hostages are held by a foreign state, or by terrorists aided by a state)
All of these all situations where a handful of hostile aircraft would cause major problems, particularly since the rules of engagement are likely to prohibit shooting them down until they have been positively identified and warned, during which time they may have got dangerously close. In Sierra Leone not everyone was happy about the presence of UK forces, including the Nigerians. What if they (or even just one or two renegade jet pilots) had decided to start harassing or engaging the helicopters and we had no means to counter them?